Integrand size = 28, antiderivative size = 87 \[ \int \frac {x^2 \left (4+x^3\right )}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )} \, dx=-\sqrt {2} \arctan \left (\frac {-\frac {x^2}{\sqrt {2}}+\frac {\sqrt {1+x^3}}{\sqrt {2}}}{x \sqrt [4]{1+x^3}}\right )-\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{1+x^3}}{x^2+\sqrt {1+x^3}}\right ) \]
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\[ \int \frac {x^2 \left (4+x^3\right )}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )} \, dx=\int \frac {x^2 \left (4+x^3\right )}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {1}{\left (1+x^3\right )^{3/4}}+\frac {x}{\left (1+x^3\right )^{3/4}}+\frac {1-x+4 x^2+x^3}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )}\right ) \, dx \\ & = -\int \frac {1}{\left (1+x^3\right )^{3/4}} \, dx+\int \frac {x}{\left (1+x^3\right )^{3/4}} \, dx+\int \frac {1-x+4 x^2+x^3}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )} \, dx \\ & = -x \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {3}{4},\frac {4}{3},-x^3\right )+\frac {1}{2} x^2 \operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {3}{4},\frac {5}{3},-x^3\right )+\int \left (\frac {1}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )}-\frac {x}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )}+\frac {4 x^2}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )}+\frac {x^3}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )}\right ) \, dx \\ & = -x \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {3}{4},\frac {4}{3},-x^3\right )+\frac {1}{2} x^2 \operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {3}{4},\frac {5}{3},-x^3\right )+4 \int \frac {x^2}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )} \, dx+\int \frac {1}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )} \, dx-\int \frac {x}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )} \, dx+\int \frac {x^3}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )} \, dx \\ \end{align*}
Time = 2.22 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.85 \[ \int \frac {x^2 \left (4+x^3\right )}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )} \, dx=-\sqrt {2} \left (\arctan \left (\frac {-x^2+\sqrt {1+x^3}}{\sqrt {2} x \sqrt [4]{1+x^3}}\right )+\text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{1+x^3}}{x^2+\sqrt {1+x^3}}\right )\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.84 (sec) , antiderivative size = 208, normalized size of antiderivative = 2.39
method | result | size |
trager | \(\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{4}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{3}-2 \left (x^{3}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{3}+2 \sqrt {x^{3}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{2}-2 \left (x^{3}+1\right )^{\frac {3}{4}} x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3}}{x^{4}+x^{3}+1}\right )+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (-\frac {2 \sqrt {x^{3}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{2}+2 \left (x^{3}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{4}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{3}-2 \left (x^{3}+1\right )^{\frac {3}{4}} x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )}{x^{4}+x^{3}+1}\right )\) | \(208\) |
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Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.21 \[ \int \frac {x^2 \left (4+x^3\right )}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )} \, dx=-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \log \left (\frac {\left (i + 1\right ) \, \sqrt {2} x + 2 \, {\left (x^{3} + 1\right )}^{\frac {1}{4}}}{x}\right ) + \left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \log \left (\frac {-\left (i - 1\right ) \, \sqrt {2} x + 2 \, {\left (x^{3} + 1\right )}^{\frac {1}{4}}}{x}\right ) - \left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \log \left (\frac {\left (i - 1\right ) \, \sqrt {2} x + 2 \, {\left (x^{3} + 1\right )}^{\frac {1}{4}}}{x}\right ) + \left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \log \left (\frac {-\left (i + 1\right ) \, \sqrt {2} x + 2 \, {\left (x^{3} + 1\right )}^{\frac {1}{4}}}{x}\right ) \]
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Timed out. \[ \int \frac {x^2 \left (4+x^3\right )}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {x^2 \left (4+x^3\right )}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )} \, dx=\int { \frac {{\left (x^{3} + 4\right )} x^{2}}{{\left (x^{4} + x^{3} + 1\right )} {\left (x^{3} + 1\right )}^{\frac {3}{4}}} \,d x } \]
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\[ \int \frac {x^2 \left (4+x^3\right )}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )} \, dx=\int { \frac {{\left (x^{3} + 4\right )} x^{2}}{{\left (x^{4} + x^{3} + 1\right )} {\left (x^{3} + 1\right )}^{\frac {3}{4}}} \,d x } \]
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Timed out. \[ \int \frac {x^2 \left (4+x^3\right )}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )} \, dx=\int \frac {x^2\,\left (x^3+4\right )}{{\left (x^3+1\right )}^{3/4}\,\left (x^4+x^3+1\right )} \,d x \]
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